Calculation of Joint Moments and Muscle Forces



Calculation of Joint Moments and Muscle Forces

Try balancing on one leg. Can you feel the muscles in your calf working to help you balance?

Now try leaning forward while still on one leg. Is this easier or more difficult than just balancing?

In order to balance on one leg, a force is generated in the calf muscles. It is possible to estimate this force using position and force data, together with some moments calculations.

Figure 1 shows a free body diagram of the foot for a gymnast balancing on one leg. The forces acting on the foot and their locations relative to the ankle joint are illustrated. Force ‘R’ is the reaction force of the ground in response to the weight of the gymnast. Force ‘mg’ is the weight of the foot, where ‘m’ is the mass of the foot and ‘g’ is the acceleration due to gravity. These data can be used to estimate the muscle moment that must be produced by the calf muscles for equilibrium to be maintained (i.e. for the gymnast to stay balanced).

Figure 1 – Forces acting on the foot segment Figure 2 – Muscle force and moments

for a gymnast balancing on one leg

If the distance from the ankle joint centre the foot centre of gravity is measured as 0.05 m and the distance from the ankle joint centre to the reaction force R is 0.20 m, then it is possible to calculate the moment about the ankle joint centre.

Taking moments about the ankle joint centre:

Ma = (0.20 x R) – (0.05 x mg)

The force R is the weight of the subject. If the mass of the gymnast is 45 kg and the mass of their foot is 0.6 kg, you can calculate the muscle moment about the ankle joint that is required for equilibrium to be maintained.

Ma = (0.20 x 450) – (0.05 x 6) = 90 + 0.3 = 90.3 N.m

This ankle joint moment is generated by the force of the calf muscles, shown as F is Figure 2. If the distance from the ankle joint centre to the force F is measured, then F can be calculated:

Ma = 90.3 = d1 x F

If d1 = 0.04 m, then F = 90.3 / 0.04 = 2257.5 N

This force can also be calculated as multiples of bodyweight: 2257.5/450 = 5.0 times bodyweight

Suggested Developments

- estimation of ankle joint moment and muscle force when the gymnast raises her heel and supports the weight on the ball of the foot

- progression to determination of ankle joint moments during running?

Further Examples

- calculation of whole body centre of gravity for different body orientations in sports, given data on the segment positions and centres of gravity

- calculation of moments of inertia during flight for different body positions

Example:

Figure 3 shows a gymnast somersaulting following a high bar dismount. Using information provided in Table 1, calculate the position of the gymnast’s centre of gravity and mark this with a cross.

Calculate the moment of inertia about the point of rotation.

Assume a body mass of 60kg, and a scale of 1mm (measured) = 0.02m (reality).

Table 1 – Segmental relative weight, centre of gravity and moment of inertia data.

Segment Relative Weight Centre of Gravity Moment of Inertia about CG (kg.m2)

Head 0.073 46.4% vertex to chin 0.0248

Trunk 0.507 38.0% shoulder to hip 1.2606

Upper Arm 0.026 51.3% shoulder to elbow 0.0213

Forearm 0.016 39.0% elbow to wrist 0.0076

Hand 0.007 82.0% wrist to knuckle 0.0005

Thigh 0.103 37.2% hip to knee 0.1052

Calf 0.043 37.1% knee to ankle 0.0504

Foot 0.015 44.9% heel to toe 0.0038

Figure 3 – Gymnast following a dismount from the high bar.

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